The pointwise densities of the Cantor measure
type: Item
instance of: scholarly article
title: The pointwise densities of the Cantor measure
zbMATH Open document ID: 0967.28006
author: De-Jun Feng
author: Zhi-Ying Wen
author: Hua Su
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-08-27T00:00:00Z
review text: Let \({\mathcal C}\) be the classical middle-third Cantor set and let \(\mu_{\mathcal C}\) be the Cantor measure. Let \(s = \log 2/ \log 3\) be the Hausdorff dimension of \({\mathcal C}\). For every \(x \in {\mathcal C}\), the authors determine explicitly the values of the upper and lower \(s\)-densities \(\Theta^{* s}(\mu_{\mathcal C}, x)\) and \(\Theta_*^{s} (\mu_{\mathcal C}, x)\) of \(\mu_{\mathcal C}\) at \(x\). In particular, they show that, for \(\mu_{\mathcal C}\)-almost all \(x\), \(\Theta^{* s}(\mu_{\mathcal C}, x)= 2\cdot 4^{-s}\) and \(\Theta_*^{s} (\mu_{\mathcal C}, x) = 4^{-s}\). As a consequence, this latter result shows that the \(s\)-dimensional packing measure of \({\mathcal C}\) is \(4^s\).
reviewed by: Y. Xiao
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1542474
zbMATH Keywords: Cantor measure
zbMATH Keywords: upper and lower density
zbMATH Keywords: packing measure
zbMATH Keywords: middle-third Cantor set
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7137
OpenAlex ID: W2121246956
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DOI: 10.1006/JMAA.2000.7137
description: scientific article; zbMATH DE number 1542474